I want to buy an off-the-shelf remote control device that has powered wheels that is cheap and that I can activate while performing a magic act on the stage. I'm referring to this as "basebot".

I bought a $39 cardboard stand up of R2D2 that I plan to mount on the basebot and bring him out with a yellow silk etc as a comedy bit for 2 minutes during my act ("my assistant is on vacation so I've hired a temporary replacement").

Since I do a sci-fi based act I may have several different cardboard standups (packs small plays big) each with its own basebot.

Initially my basebot (with R2D2) will be in the wings of the stage so that at the appropriate time when I push a button he will come out onto the stage and a minute later when I push a button a second time he will return to the wings. This means I do not need any steering controls since he will just come straight on and go straight back (I could glue the steering mechanism in place).

This is a one-man act so I do not want to rely on any off stage assistance.

I will have a four-wheeled free-wheeling base that the off the shelf robot will be velcro attached to. The cardboard stand-up will be mounted to the free-wheeling base.

Below are my ideal criteria for my basebot (I know that these criteria will probably not all be met):
1. push of a button brings him on.
2. non-infrared remote (need a reliable remote)
3. powerful enough to move the cardboard stand-up R2D2.
4. needs to be slow (most RC toys are way too fast except for the proportional speed controlled crawlers but they can be expensive)
5. cheap under $50 (but I may have to go higher)
6. single drive control (tank controls are bad for this because it requires two hands also the timing when pushing the two levers has to be perfect to make it go straight)
7. no "time outs" (most programmable toy robots have either a fixed three minute or fixed five minute time out shut down when not in use, this is very bad for my purposes)
8. not too noisy
9. two-wheel drive goes straighter than four-wheel drive

I have bought, tested, returned a bunch of toys and they are all weak in one or more of my criteria:
a. fetch robot (has 2 or 3 minutes time out)
b. botley robot (has 5 minutes time out)
c. rad robot (3 minute time out, 5 minute in stealth mode)
d. Target Jurassic World RC Jeep (almost slow enough but to noisy in reverse)
e. Hobby Lobby big RC crawler (plenty powerful but way too fast for my needs, it needs proportional speed control)
f. Hobby Lobby blue RC Jeep crawler/off road vehicle (this may work because it has reliable 2.4 remote and is almost slow enough and is two-wheel drive which is good). I wish it had proportional speed control.
g. Walmart grave digger crawler (same comments as the Hobby Lobby Jeep)
h. maisto r/c rock crawler from Amazon (this meets all my criteria except it is way too fast for my needs).

I have not found any toy grade RC vehicles that have proportional speed control.

Note that to get proportional speed control you normally need to buy a hobby grade RC vehicle and they typically cost more than $100.

That being said I found and have on order a 2 wheel drive (a good thing) WPL D12 RC delivery truck that has proportional speed control and cost around $50.

I have looked at the numerous programmable Arduino and raspberry pie robot base kits but I want something that's off the shelf ready to run because I've got way too many other projects going on to spend time on a kit.

If any of you out there know of another solution to my criteria please post or if you know of a toy grade RC that has proportional speed control.
Thanks.

Additional note: I used a programmable small 10-in high R2D2 toy to power my stand up cardboard R2D2 in an actual show at the police youth camp and it did work just barely.

The problems were that the little programmable R2D2 toy does not have a reverse so you have to program it to turn around which I did and it worked but the new aiming for its return can be a problem also it is just barely powerful enough to do the job so I held my breath during the show and it happened to work but I need something better.

Note that a programmable toy solution is better than an RC solution because you can push a button and do other things during the 5 seconds that it takes the robot to come on stage whereas an RC toy you have the hold the trigger the entire 5 seconds while it's coming on and same when the robot is going off.

On the flip side the RC toys seem to be more robust.

This is not a showstopper, either way is acceptable.

What I'm about to say probably comes with a lot less "direct expertise" than what it sounds like you're looking for. Nevertheless the source I'm suggesting isn't on your list so you might want to know about it. If you go to aliexpress.com and type in "robot toy" you will not only get a copious number of suggestions, many of the pricepoints are way lower than your $50. Now this site is really enormous, and in truth if what you really wanted to do was to build a mostly custom robot you can also use this site as the source for the components you might need, including motors (all kinds of DC technologies - torque, BLDC, stepper), motor drivers, shafts and shaft couplers, bearings, batteries, timing belts and pulleys, wheels, tracks, chassis, optical encoders, and no limit of electronic assemblies up to and including Raspberry Pi and Arduino microcomputers. So at some level it's sort of a "limitless source" of these sorts of supplies. Unfortunately I can't help you with questions like "which preassembled robot toy for sale here has the most torque", but sometimes you can tell things from context (for example you'd think something with tracks probably has more torque than one with wheels) and some have partial data sheets, but at least it's an interesting collection of choices to look at, and might even help you develop your sense of what you're designing. And hey at least it's more choices than you had!

Thanks for the heads up I will definitely look at their website!